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Topology : Connected Spaces and Explosion Point

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B6. (a) Define what it means for a topological space to be connected.
(b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.
(c) Suppose that X is connected and that A is a connected subset of X. Suppose further that X ? A = U U V, where U and V are nonempty disjoint open subsets of X ? A in the relative topology for X ? A. Show that A U U is connected.
(d) (i) Define what is meant by a component of a topological space.
(ii) A space X is totally disconnected if and only if each of its components consists of a point. Show that Q in the relative topology as a subset of R with the usual topology is totally disconnected.
(iii) A connected space X is said to have an explosion point p if and only if X ? {p} is totally disconnected, Find an example of such a space.

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Solution Preview

Please see the attachment.
For the last question about the example of a connected space with an explosion point, I only find a specific small example. I can not find a general example.


a. A topological space is connected if it can not be expressed as the union of two nonempty disjoint open sets.
b. Proof:
Since is open in and is a subspace in , then we can find an open set in , such that . Similarly, we can find an open set , such that . ...

Solution Summary

Connected Spaces and an Explosion Point are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.